A circle is described around a regular hexagon and a circle is inscribed in it. The length of the larger circle

A circle is described around a regular hexagon and a circle is inscribed in it. The length of the larger circle is 4pi Find the area of the ring, the area and perimeter of the hexagon.

1. It is known that the circumference is equal to 2 P * R. According to the problem statement, it is given: the length of the circumscribed circle is 4 P, so R = 4 P: 2 P = 2.

2. The center of the circumscribed and hence the inscribed circle is connected to each vertex of a regular hexagon.

Received six isosceles triangles, each of which has a side equal to R, and the angle at the apex is 360 °: 6 = 60 °, that is, all triangles are equilateral and then the base is equal to R = 2.

Let’s calculate the value of the perimeter:

P = 2 * 6 = 12.

3. From the center of the circles draw the heights h to the sides of the hexagon, which divide each side in half. We calculate the value of h by the Pythagorean theorem:

h = √R² – (2/2) ² = √4 – 1 = √3, and hence the radius of the inscribed circle r = √3.

4. Let’s calculate the area

a) the circumscribed circle П * R² = П * 2² = 4 П;

b) inscribed P * r² = P * √3² = 3 P.

c) rings S = 4 P – 3 P = P.

d) hexagon 1/2 * side * r * 6 = 1/2 * 2 * √3 * 6 = 6 √3.